平坦广群的复杂性
The complexity of a flat groupoid
摘要：
Grothendieck证明了任何有限满射的诺特计划的因素，成为一个有限序列的有效满射。我们定义具有有限稳定子的平坦广群$R\rightrightarrows X$的复杂度为有限映射$R\to X\times\_{X/R} X$的标准序列的长度，其中$X/R$是Keel-Mori几何商.对复杂度不超过1的广群，证明了一个沿商$X\to X/R$的下降定理和一个广群与正规子群的商定理。我们期望这种复杂性能在群胚的子群的更精细研究中发挥重要作用。
英文摘要：
Grothendieck proved that any finite epimorphism of noetherian schemes factors into a finite sequence of effective epimorphisms. We define the complexity of a flat groupoid $R\rightrightarrows X$ with finite stabilizer to be the length of the canonical sequence of the finite map $R\to X\times\_{X/R} X$, where $X/R$ is the Keel-Mori geometric quotient. For groupoids of complexity at most 1, we prove a theorem of descent along the quotient $X\to X/R$ and a theorem of quotient of a groupoid by a normal subgroupoid. We expect that the complexity could play an important role in the finer study of quotients by groupoids.
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